Exercise 3: Manually Graphing the Projection of the Trefoil Knot onto the xy-Plane

In this video I manually (and painstakingly) plot an approximation of the projection of the 3D trefoil knot onto the xy-plane. I begin by converting the parametric equations of the trefoil knot into polar coordinates, and then find the values where it is max and minimum until the projection repeats itself. I then determine the min and max of the z values, and use this to determine where the projection overlaps at each intersection point. I then compare my manual graph with the computer-drawn one, and my approximation leaves out the 3 inflection points. I also plot the trefoil knot in 3D using the amazing GeoGebra graphing calculator as well as the new Desmos' graphing calculator.

#math #trefoilknot #geogebra #desmos #calculus

Timestamps

• Exercise 3: Trefoil Knot: 0:00
• Solution: Projection of the curve onto the xy-plane: 2:41
• Converting to polar coordinates: 3:27
• Polar equation for the projection onto the xy-plane: 10:37
• Obtaining minimum and maximum values of r for various angles until we rotate back to the starting point: 11:11
• Manually graphing an approximate projection on the xy-plane: 23:09
• Determining how the curve overlaps by finding the minimum and maximum values of z. Initially finding max values: 28:52
  • Minimum values for z: 35:45
  • Minimum and maximum values for z occurs at r = 2: 38:50
• Comparing max and min values of z at intersection points to determine overlap: 41:06
• Computer-drawn graphs of the Trefoil knot and its projection, which has inflection points: 45:39
• Graphing the trefoil knot in GeoGebra: https://www.geogebra.org/calculator/ckmexx42 48:15
  • Using Grok AI to plot a tube around the trefoil knot: https://x.com/i/grok/share/TPVALzOLEOm5h4GLZhx6OuBK5 48:35
  • Plotting in Desmos' new 3D graphing calculator: https://www.desmos.com/3d/dn5bonsvwf 49:41

Notes and playlists

• Summary: @mes/re-leothreads-34k6ldabs
• Sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0GwdMY-E9_4LsmYmO-IxcoX
• Notes: @mes/omhbcxkw
• Vector Functions playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HQl7xTQBS_O8Te8PcpNm4U .

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